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I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understand

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion about that specific part of course material. If you don't understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality. Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary. I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts. Especially when you start to study mathematics, the first couple semesters are foundational and if you don't understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in. I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.

benrbray 6y ago
I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.
- kkylin 6y ago
  Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with the motivation (e.g., electrical circuits or statistical regression or what have you) later. This lack of context makes it very hard to see how things fit together. For my own part, I usually learned the material from one course when I needed to use it in a subsequent course -- it's in going back to review the material when I really learn it. Edit: also want to add that providing context in early courses isn't always easy, because students are much more diverse now in their career plans. It used to be (this tells you how old I am) that many of the students taking calculus would also be taking (or have taken) a calculus-based physics course. Those days are long gone.
  - johnsonjo 6y ago
    I took Physics as an elective in University, but it really didn’t sharpen my Calculus skills. Which is sad, but probably because I did so bad in Calculus as an undergraduate. But, one thing I absolutely did take away from Physics was the idea of units and their conversions. It has come in handy so often though in and outside my degree of computer science. If I had to simplify it, it really helps you not to compare apples to oranges. At the beginning of Street Fighting Mathematics it brings up that point. For example it isn’t a good comparison to compare the wealth in assets of a given company to a countries GDP. Because, GDP is happening over every year in time whereas the wealth is merely effected by what point in time it’s observed. Units help one see that in order to be truly and more fairly comparable they have to be the same type of units. I used an analogy of units in physics recently to help a teammate with types and generics, sadly I can’t remember exactly what I said haha other than don’t compare apples to oranges, but I think I had further insight than just that.
    - kkylin 6y ago
      Absolutely! internalizing dimensional analysis has been one of the most useful things I learned in grad school -- even though converting units was something many of us learn, in one form or another, before college.
      - lmilcin 6y ago
        It is interesting how some of the things just feel like common knowledge when they are not. I frequently see people comparing things that aren't really comparable and it always seems to me it should be obvious this can't be done. Most common example is mixing watts and watt*hours. Which should be pretty obvious. Nobody mixes by accident the speed at which you travel with the distance you have to travel. Nobody says "My car can pull off 200kms" or "I have 120km/h to cover" yet the same happens for energy vs power on a daily basis, even in publications.
  - ghaff 6y ago
    >It used to be (this tells you how old I am) Why do you say that? Maybe it's atypical but MIT seems to have the same calculus and physics sequence, taken in parallel, that they did ages ago.
- hawkice 6y ago
  > students really do require an experienced guide to navigate through the tangled web of concepts. Consider how much people pay for education and don't get this, and how much some people are paid to be teachers who do not provide it.
  - benrbray 6y ago
    Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem. The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!
    - wolfram74 6y ago
      I'm now interested in a list of transactions ordered by participant dissatisfaction, like, paying for gym memberships and not getting the outcomes they want, therapy or medical attention to mostly be ignored by the provider, ISPs in the US, online shopping to get something wildly different. Is education especially bad compared to these things? or just that it's such a large expenditure we just-world-fallacy ourselves into thinking it ought to work better?
      - spijdar 6y ago
        Is it really a logical fallacy to assume that we should have better outcomes from spending tens to hundreds of thousands of dollars on academia vs e.g. spending thirty dollars on amazon? At least naively, that money could (should?) be going towards paying for staff and a support system/hierarchy to ensure a higher quality. That's at least nominally why accredited, non-profit education is so expensive, because it's nominally supposed to be capable of paying decent salaries to intelligent people dedicated to teaching, compared to say amazon or the gym where the incentives are to cost cut and maximize profit. In reality, I suspect the same applies to colleges, but I don't think it should, in some ideal world.
    - abdullahkhalids 6y ago
dzolob 6y ago
There is no absolute here. While I agree that a great degree of understanding is desirable, lots of concepts and techniques don’t sink in until hours, days or even years after being presented. I still remember powering through lots problems without much clarity and achieving the right answer. Every attempt, failed or successful, got me closer to really understand what I was doing. It’s ok not to understand a topic the first or nth-time time you come across it. If you follow through, something will stick, and sooner than later you’ll get a good enough picture. They are like seeds, you know.
- lmilcin 6y ago
  I guess what you mean by "sink in" is a form of deeper more intuitive knowledge. Like when you suddenly figure out how everything is connected. If that's what you mean, then yes -- this is something that is going to happen over time. What I mean is that your teacher will want you to understand some concepts and learn some skills before you move to next topic. Very deep understanding is probably not on the list and it is expected to only form over time. But there are topics that are explicitly expected and you have to understand because otherwise what is going to be on the board the next day is not going to make whole lot of sense.
  - dzolob 6y ago
    Yes and no. What I meant was more like, you don’t need to fully understand rigid body mechanics to build a table, but if you build enough tables, you might win some knowledge on rigid body mechanics that you may have missed in class. Of course, this is not always true, but a concrete example that comes to mind is integration. It’s very hard conceptually, but not so in technique. If you learn how to do it, for the most part you can skip the concept (to some degree, obviously). I didn’t understood Riemann integral until I learned Lebesque.
JJMcJ 6y ago
Mathematics is high on the list of "you can't fall behind and cram". This is true not only within a single class but from class to class. Aren't fluent in manipulating equations? Calculus is going to be misery. Don't know calculus like the layout of your bedroom? Differential equations aren't going to happen. All the way back to `2+3 = 5` in first grade. If that doesn't go well, the rest will likely not, either.
milesvp 6y ago
> the first couple semesters are foundational and if you don't understand something it is like a pyramid I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the precision necessary to use English to prove something mathematically. It wasn't until after graduation that I learned that there are some sophomore math courses at my university that Math majors usually take that were unofficially known as "intro to proofs". I have no doubt this was missing foundation that made 400 level math courses so painful for me.
- grayclhn 6y ago
  Those upper division math classes are pretty painful regardless. The material is just innately hard in a way that's dissimilar to most other hard things.
grayclhn 6y ago
I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there's no amount of "understanding" that's going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in. I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.
bmitc 6y ago
The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater. I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions. A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting. One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.
- hintymad 6y ago
  I thought this was what university did. For instance, there's a little stats in calculus courses, then there's stats in mathematical stats courses, then in queuing theory course, then in networking course, and of course in all kinds of statistical learning and machine learning courses. By the time you finish those courses, you will get a pretty good grasp of basic stats.
  - lmilcin 6y ago
    As long as we are talking math, you can take a bunch of books and learn the same without need for university. You can even go to talk to somebody who will tell you what books to take and in what order. What university does is help you connect with other people who study the same, to observe or work with them when solving problems and to have access to people working the cutting edge of the discipline you are studying who you would not normally have access to. Also, it gives a little bit pressure which might be important for some or most people. Oh, and official credentials.
BrandoElFollito 6y ago
I agree with this as the father of two kids in high school. I routinely force them to fully understand some things they learn, and I am more liberal with others. Mathematics is one of these subjects where there is almost nothing you can afford not to understand. It will absolutely bite you someday. When learning math there are things that are really new (such as differentials, or operations on fractions) that are conceptually different from other things you just learn (Pythagoras theorem for instance) and I spend a lot of time with them to help them understand the "why" of these operations. Otherwise i know they will have a very, very hard time to follow up. Physics is lighter - you can have trouble understanding thermodynamics and it will not mean much when doing mechanics. There are parts of physics I never really understood despite having a PhD in physics. This is in sharp contrast with, say, history where missing a bit does not impact the next things one learns (at least at high school level - and this is from someone who loves history).
hnracer 6y ago
I did well in my college math exams (top 5 percent of class) by just memorizing proofs and memorizing the methods to solve common problems without understanding much of anything about what I was doing. The exams were very much hackable. I wouldn't be able to pull this off in grad level maths but undergrad it's feasible. It was a total waste of time though and I regret that approach, but I did what I was given an incentive to do (least effort for max reward)
hintymad 6y ago
Maybe college-level math is different from math taught in a maths depart? Either way, the common errors mentioned in the article appear to be, well, elementary. I'd imagine 6 or 7 graders make those mistakes, but college students? Shouldn't they make mistakes like "there is no function that is everywhere continuous but nowhere differentiable", or "isomorphism of factors implies isomorphism of quotient groups"? The webpage says a lot about the quality of the K12 education in the US.

benrbray 6y ago

I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.

kkylin 6y ago
Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with the motivation (e.g., electrical circuits or statistical regression or what have you) later. This lack of context makes it very hard to see how things fit together. For my own part, I usually learned the material from one course when I needed to use it in a subsequent course -- it's in going back to review the material when I really learn it. Edit: also want to add that providing context in early courses isn't always easy, because students are much more diverse now in their career plans. It used to be (this tells you how old I am) that many of the students taking calculus would also be taking (or have taken) a calculus-based physics course. Those days are long gone.
- johnsonjo 6y ago
  I took Physics as an elective in University, but it really didn’t sharpen my Calculus skills. Which is sad, but probably because I did so bad in Calculus as an undergraduate. But, one thing I absolutely did take away from Physics was the idea of units and their conversions. It has come in handy so often though in and outside my degree of computer science. If I had to simplify it, it really helps you not to compare apples to oranges. At the beginning of Street Fighting Mathematics it brings up that point. For example it isn’t a good comparison to compare the wealth in assets of a given company to a countries GDP. Because, GDP is happening over every year in time whereas the wealth is merely effected by what point in time it’s observed. Units help one see that in order to be truly and more fairly comparable they have to be the same type of units. I used an analogy of units in physics recently to help a teammate with types and generics, sadly I can’t remember exactly what I said haha other than don’t compare apples to oranges, but I think I had further insight than just that.
  - kkylin 6y ago
    Absolutely! internalizing dimensional analysis has been one of the most useful things I learned in grad school -- even though converting units was something many of us learn, in one form or another, before college.
    - lmilcin 6y ago
      It is interesting how some of the things just feel like common knowledge when they are not. I frequently see people comparing things that aren't really comparable and it always seems to me it should be obvious this can't be done. Most common example is mixing watts and watt*hours. Which should be pretty obvious. Nobody mixes by accident the speed at which you travel with the distance you have to travel. Nobody says "My car can pull off 200kms" or "I have 120km/h to cover" yet the same happens for energy vs power on a daily basis, even in publications.
- ghaff 6y ago
  >It used to be (this tells you how old I am) Why do you say that? Maybe it's atypical but MIT seems to have the same calculus and physics sequence, taken in parallel, that they did ages ago.
hawkice 6y ago
> students really do require an experienced guide to navigate through the tangled web of concepts. Consider how much people pay for education and don't get this, and how much some people are paid to be teachers who do not provide it.
- benrbray 6y ago
  Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem. The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!
  - wolfram74 6y ago
    I'm now interested in a list of transactions ordered by participant dissatisfaction, like, paying for gym memberships and not getting the outcomes they want, therapy or medical attention to mostly be ignored by the provider, ISPs in the US, online shopping to get something wildly different. Is education especially bad compared to these things? or just that it's such a large expenditure we just-world-fallacy ourselves into thinking it ought to work better?
    - spijdar 6y ago
      Is it really a logical fallacy to assume that we should have better outcomes from spending tens to hundreds of thousands of dollars on academia vs e.g. spending thirty dollars on amazon? At least naively, that money could (should?) be going towards paying for staff and a support system/hierarchy to ensure a higher quality. That's at least nominally why accredited, non-profit education is so expensive, because it's nominally supposed to be capable of paying decent salaries to intelligent people dedicated to teaching, compared to say amazon or the gym where the incentives are to cost cut and maximize profit. In reality, I suspect the same applies to colleges, but I don't think it should, in some ideal world.
  - abdullahkhalids 6y ago

dzolob 6y ago

There is no absolute here. While I agree that a great degree of understanding is desirable, lots of concepts and techniques don’t sink in until hours, days or even years after being presented. I still remember powering through lots problems without much clarity and achieving the right answer. Every attempt, failed or successful, got me closer to really understand what I was doing. It’s ok not to understand a topic the first or nth-time time you come across it. If you follow through, something will stick, and sooner than later you’ll get a good enough picture. They are like seeds, you know.

lmilcin 6y ago
I guess what you mean by "sink in" is a form of deeper more intuitive knowledge. Like when you suddenly figure out how everything is connected. If that's what you mean, then yes -- this is something that is going to happen over time. What I mean is that your teacher will want you to understand some concepts and learn some skills before you move to next topic. Very deep understanding is probably not on the list and it is expected to only form over time. But there are topics that are explicitly expected and you have to understand because otherwise what is going to be on the board the next day is not going to make whole lot of sense.
- dzolob 6y ago
  Yes and no. What I meant was more like, you don’t need to fully understand rigid body mechanics to build a table, but if you build enough tables, you might win some knowledge on rigid body mechanics that you may have missed in class. Of course, this is not always true, but a concrete example that comes to mind is integration. It’s very hard conceptually, but not so in technique. If you learn how to do it, for the most part you can skip the concept (to some degree, obviously). I didn’t understood Riemann integral until I learned Lebesque.

JJMcJ 6y ago

Mathematics is high on the list of "you can't fall behind and cram". This is true not only within a single class but from class to class. Aren't fluent in manipulating equations? Calculus is going to be misery. Don't know calculus like the layout of your bedroom? Differential equations aren't going to happen. All the way back to `2+3 = 5` in first grade. If that doesn't go well, the rest will likely not, either.

milesvp 6y ago

> the first couple semesters are foundational and if you don't understand something it is like a pyramid I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the precision necessary to use English to prove something mathematically. It wasn't until after graduation that I learned that there are some sophomore math courses at my university that Math majors usually take that were unofficially known as "intro to proofs". I have no doubt this was missing foundation that made 400 level math courses so painful for me.

grayclhn 6y ago
Those upper division math classes are pretty painful regardless. The material is just innately hard in a way that's dissimilar to most other hard things.

grayclhn 6y ago

I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there's no amount of "understanding" that's going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in. I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.

bmitc 6y ago

The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater. I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions. A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting. One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.

hintymad 6y ago
I thought this was what university did. For instance, there's a little stats in calculus courses, then there's stats in mathematical stats courses, then in queuing theory course, then in networking course, and of course in all kinds of statistical learning and machine learning courses. By the time you finish those courses, you will get a pretty good grasp of basic stats.
- lmilcin 6y ago
  As long as we are talking math, you can take a bunch of books and learn the same without need for university. You can even go to talk to somebody who will tell you what books to take and in what order. What university does is help you connect with other people who study the same, to observe or work with them when solving problems and to have access to people working the cutting edge of the discipline you are studying who you would not normally have access to. Also, it gives a little bit pressure which might be important for some or most people. Oh, and official credentials.

BrandoElFollito 6y ago

I agree with this as the father of two kids in high school. I routinely force them to fully understand some things they learn, and I am more liberal with others. Mathematics is one of these subjects where there is almost nothing you can afford not to understand. It will absolutely bite you someday. When learning math there are things that are really new (such as differentials, or operations on fractions) that are conceptually different from other things you just learn (Pythagoras theorem for instance) and I spend a lot of time with them to help them understand the "why" of these operations. Otherwise i know they will have a very, very hard time to follow up. Physics is lighter - you can have trouble understanding thermodynamics and it will not mean much when doing mechanics. There are parts of physics I never really understood despite having a PhD in physics. This is in sharp contrast with, say, history where missing a bit does not impact the next things one learns (at least at high school level - and this is from someone who loves history).

hnracer 6y ago

I did well in my college math exams (top 5 percent of class) by just memorizing proofs and memorizing the methods to solve common problems without understanding much of anything about what I was doing. The exams were very much hackable. I wouldn't be able to pull this off in grad level maths but undergrad it's feasible. It was a total waste of time though and I regret that approach, but I did what I was given an incentive to do (least effort for max reward)

hintymad 6y ago

Maybe college-level math is different from math taught in a maths depart? Either way, the common errors mentioned in the article appear to be, well, elementary. I'd imagine 6 or 7 graders make those mistakes, but college students? Shouldn't they make mistakes like "there is no function that is everywhere continuous but nowhere differentiable", or "isomorphism of factors implies isomorphism of quotient groups"? The webpage says a lot about the quality of the K12 education in the US.